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Encyclopedia of Complexity and Systems Science pp 243–257Cite as

Algorithmic Complexity and Cellular Automata

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Definition of the Subject

In the last 10 years the field of complex systems has enjoyed astonishing development withmeaningful applications in most scientific domains.

Cellular automata (CA) are a very used model for complex systems characterized by a multitude of smallidentical agents capable of building a complex behavior on the basis of local interactions.

The huge variety of CA dynamical behaviors has been popularized since the early 1980s by the workof S. Wolfram (see [18] for an exhaustive review).

Later, researchers started a systematic study of CA. Most of the results are summarized in the chaptersof this book (see Chaotic Behavior of Cellular Automata, Dynamics of Cellular Automata in Non-compact Spaces, Topological Dynamics of Cellular Automata and Ergodic Theory of Cellular Automata for instance).

However, many questions seem to be intractable and vanquisch researchers' efforts (some such efforts are reported by Chaotic Behavior of Cellular Automata, Dynamics of Cellular...

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Abbreviations

Algorithmic complexity of object x :

shortest program for outputting a description for x(w.r.t. a universal representation system)

Equicontinuity:

all points are equicontinuity points (in compact settings)

Equicontinuity point:

a point for which the orbits of nearby points remain close

Expansivity:

from two distinct points orbits eventually separate

Incompressible word:

a word for which the shortest program outputting it has “almost” thesame length as the word itself

Injectivity:

the next state function is injective

Kolmogorov complexity:

see “algorithmic complexity”

Rich configuration:

a configuration that contains all possible finite patterns overa given alphabet

Sensitivity to initial conditions:

for any point x there exist arbitrary close pointswhose orbits eventually separate from the orbit of x

Surjectivity:

the next state function is surjective

Transitivity:

there always exist points that eventually move from any arbitraryneighborhood to any other

Bibliography

Primary Literature

  1. Blanchard F, Formenti E, Kurka P (1999) Cellular automata in the Cantor, Besicovitch andWeyl topological spaces. Complex Syst 11:107–123

    MathSciNet  Google Scholar 

  2. Blanchard F, Cervelle J, Formenti E (2003) Periodicity and transitivity for cellularautomata in Besicovitch topologies. In: Rovan B, Vojtas P (eds) (MFCS'2003), vol 2747. Springer,Bratislava, pp 228–238

    Google Scholar 

  3. Blanchard F, Cervelle J, Formenti E (2005) Some results about chaotic behavior ofcellular automata. Theor Comput Sci 349(3):318–336

    MathSciNet  MATH  Google Scholar 

  4. Brudno AA (1978) The complexity of the trajectories of a dynamical system. Russ MathSurv 33(1):197–198

    MathSciNet  MATH  Google Scholar 

  5. Brudno AA (1983) Entropy and the complexity of the trajectories of a dynamical system.Trans Moscow Math Soc 44:127

    Google Scholar 

  6. Calude CS, Hertling P, Jürgensen H, Weihrauch K (2001)Randomness on full shift spaces. Chaos, Solitons Fractals 12(3):491–503

    Google Scholar 

  7. Cattaneo G, Formenti E, Margara L, Mazoyer J (1997) A shift‐invariant metric onS Z inducing a non‐trivial topology. In: Privara I, Rusika P (eds) (MFCS'97), vol 1295.Springer, Bratislava, pp 179–188

    Google Scholar 

  8. Cervelle J, Durand B, Formenti E (2001) Algorithmic information theory and cellularautomata dynamics. In: Mathematical Foundations of Computer Science (MFCS'01). Lectures Notes inComputer Science, vol 2136. Springer, Berlinpp 248–259

    Google Scholar 

  9. Delorme M, Mazoyer J (1999) Cellular automata as languages recognizers. In: Cellularautomata: A parallel model. Kluwer, Dordrecht

    Google Scholar 

  10. Durand B, Levin L, Shen A (2001) Complex tilings. In: STOC '01: Proceedings of the33rd annual ACM symposium on theory of computing, pp 732–739

    Google Scholar 

  11. Kari J (1994) Rice's theorem for the limit set of cellular automata. Theor Comput Sci127(2):229–254

    MathSciNet  MATH  Google Scholar 

  12. Kůrka P (1997) Languages, equicontinuity and attractors in cellular automata.Ergod Theory Dyn Syst 17:417–433

    Google Scholar 

  13. Li M, Vitányi P (1997) An introduction to Kolmogorov complexity and its applications,2nd edn. Springer, Berlin

    Google Scholar 

  14. M Delorme EF, Mazoyer J (2000) Open problems. Research Report LIP 2000-25, EcoleNormale Supérieure de Lyon

    Google Scholar 

  15. Pivato M (2005) Cellular automata vs. quasisturmian systems. Ergod Theory Dyn Syst25(5):1583–1632

    MathSciNet  MATH  Google Scholar 

  16. Robinson RM (1971) Undecidability and nonperiodicity for tilings of the plane. InventMath 12(3):177–209

    MathSciNet  ADS  Google Scholar 

  17. Terrier V (1996) Language not recognizable in real time by one-way cellular automata.Theor Comput Sci 156:283–287

    MathSciNet  Google Scholar 

  18. Wolfram S (2002) A new kind of science. Wolfram Media, Champaign.http://www.wolframscience.com/

Books and Reviews

  1. Batterman RW, White HS (1996) Chaos and algorithmic complexity. Fund Phys26(3):307–336

    MathSciNet  ADS  Google Scholar 

  2. Bennet CH, Gács P, Li M, Vitányi P, Zurek W (1998) Information distance. EEE Trans InfTheory 44(4):1407–1423

    Google Scholar 

  3. Calude CS (2002) Information and randomness. Texts in theoretical computer science, 2ndedn. Springer, Berlin

    Google Scholar 

  4. Cover TM, Thomas JA (2006) Elements of information theory, 2nd edn.Wiley, New York

    MATH  Google Scholar 

  5. White HS (1993) Algorithmic complexity of points in dynamical systems. Ergod Theory DynSyst 13:807–830

    MATH  Google Scholar 

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Acknowledgments

This work has been partially supported by the ANR Blanc Project “Sycomore”.

Author information

Authors and Affiliations

  1. Laboratoire d'Informatique de l'Institut Gaspard–Monge, Université Paris-Est, Marne la Vallée, France

    Julien Cervelle

  2. Laboratoire I3S, Université de Nice-Sophia Antipolis, Sophia Antipolis, France

    Enrico Formenti

Authors
  1. Julien Cervelle
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  2. Enrico Formenti
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Editor information

Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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© 2009 Springer-Verlag

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Cervelle, J., Formenti, E. (2009). Algorithmic Complexity and Cellular Automata. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_17

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